An Introduction to Matrix Lie Groups

ABOUT THE COURSE:This course offers an introduction to the theory of matrix Lie groups which are groups of invertible matrices that act as geometric transformations of space. We explore their geometric structures, focusing on key examples of matrix groups. Emphasis is placed on the interplay between group theory, geometry, and linear algebra. The course is designed to build foundational tools for further study in Lie groups, differential geometry, and mathematical physics.The course assumes minimal prerequisites and should be accessible to anyone with basic knowledge of group theory, linear algebra, and calculus.PREREQUISITES: Calculus, basic linear algebra and basic group theory. Basic knowledge on general topology can be useful, but not necessary.INDUSTRY SUPPORT:Any industry that deal with Quantum computations, Robotics and Control theory should recognize this course.

Syllabus

Week 1: Geometry of complex numbers. The quaternions. Motivating examples: Space rotations and Rotations of a sphere.
Week 2:The general linear groups. Conjugation and Change of basis. . All matrix groups are real matrix groups.
Week 3:The Unitary groups. The Euclidean isometry group. Low dimensional examples.
Week 4:Topology of matrix groups. Open sets. Continuity. Connected sets. Compact sets.
Week 5:Lie algebras. Examples. Lie algebras as vector fields. The Lie algebras of orthogonal groups
Week 6:Matrix Exponentiation. Properties of Matrix Exponentiation. One parameter subgroups
Week 7:Analysis background. Differentiation. Chain rules. Inverse and Implicit function theorems.
Week 8:Restriction of exponential map to Lie algebras. Realization of matrix groups as smooth manifolds
Week 9:The Lie bracket. Adjoint representation. Example of Adjoint representation
Week 10:The double cover Sp(1) → SO(3), Some other double covers. Sketch of the Lie group-Lie algebra correspondence.
Week 11:Maximal torus. Center of compact matrix groups. Conjugates of maximal tori
Week 12:Introduction to smooth manifolds and Lie groups. Example that not all Lie groups are matrix groups. Applications of Lie groups in Natural sciences.